In the following section we define functions on adelic subspaces that give us the size of cohomology spaces as appropriate adelic integrals. We see that the Riemann-Roch theorem holds and deduce Serre's duality using Fourier analysis on number fields. In the section 4 we introduce the relative theory. In the last section we show that in our setting we obtain the Riemann-Roch theorem as stated in [1]}. In the appendix we quickly outline Fourier analysis on local non-archimedean fields that is used multiple times to complete calculations presented in this paper and is less classical than Fourier analysis on archimedean local fields.
